Just 8,162 were present at the Ricoh Arena to see two early goals in as many first-half minutes put the hosts firmly in control.
Freddy Eastwood broke the deadlock in the 15th minute with his third goal in three starts since the turn of the year.
And the Sky Blues looked to have given Palace too much of a mountain to climb just 122 seconds later as Carl Baker found the net with a sublime volley at the back post.
But Neil Danns' sixth goal of the season in the 81st minute set up a frantic finale which saw the Eagles midfielder presented with a golden opportunity to rescue a replay from the spot almost immediately from the restart - Sky Blues debutant Stephen O'Halloran had brought down Wilfried Zahabut - but his weak penalty was saved by Westwood.
Danns then missed an even easier chance with his head from two yards as Coventry just about closed out victory to secure their first triumph in seven games.